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Theorem eqvrelsymb 36705
Description: An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised and distinct variable conditions removed by Peter Mazsa, 2-Jun-2019.)
Hypothesis
Ref Expression
eqvrelsymb.1 (𝜑 → EqvRel 𝑅)
Assertion
Ref Expression
eqvrelsymb (𝜑 → (𝐴𝑅𝐵𝐵𝑅𝐴))

Proof of Theorem eqvrelsymb
StepHypRef Expression
1 eqvrelsymb.1 . . . 4 (𝜑 → EqvRel 𝑅)
21adantr 481 . . 3 ((𝜑𝐴𝑅𝐵) → EqvRel 𝑅)
3 simpr 485 . . 3 ((𝜑𝐴𝑅𝐵) → 𝐴𝑅𝐵)
42, 3eqvrelsym 36704 . 2 ((𝜑𝐴𝑅𝐵) → 𝐵𝑅𝐴)
51adantr 481 . . 3 ((𝜑𝐵𝑅𝐴) → EqvRel 𝑅)
6 simpr 485 . . 3 ((𝜑𝐵𝑅𝐴) → 𝐵𝑅𝐴)
75, 6eqvrelsym 36704 . 2 ((𝜑𝐵𝑅𝐴) → 𝐴𝑅𝐵)
84, 7impbida 798 1 (𝜑 → (𝐴𝑅𝐵𝐵𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   class class class wbr 5074   EqvRel weqvrel 36336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709  ax-sep 5222  ax-nul 5229  ax-pr 5351
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3432  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-br 5075  df-opab 5137  df-id 5485  df-xp 5591  df-rel 5592  df-cnv 5593  df-co 5594  df-dm 5595  df-rn 5596  df-res 5597  df-refrel 36616  df-symrel 36644  df-trrel 36674  df-eqvrel 36684
This theorem is referenced by:  eqvrelth  36710
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